Studying is one of the activities that causes the most stress for human beings, add to that the fact that you don’t like everything and not everything is easy, the result? A great headache, insomnia and many hours of academic work to try to understand what is going on that does not fit in your head.
We know that one of the pure sciences that causes the most headaches is mathematics, which can be as elementary as it is complex, tell that to any engineering, physics or pure mathematics student at the university level.
And within mathematics we have an area that is especially the nightmare of thousands of students a year: calculus, and more specifically, derivatives, but don’t worry, we have the solution: the chain rule.
In this guide we will teach you and give you all the tools (yes, from home!) that you need to be able to face derivatives and apply the chain rule as if you had learned and applied it since you were born.
Without further ado… let’s get started!
What do you need:
To have reached this point means that you are interested in the world of mathematics or at least you work with something or study something that is related to it. We tell you this because before investigating the chain rule, you should have learned to derive what it means that you have a predilection -at least- for science.
Let’s see everything you need to know to master the chain rule:
*We have decided to divide it into several categories to facilitate the reading and understanding of the guide.
Theoretical knowledge:
- Have at least basic knowledge or notions of calculation.
- You must know what is a function and what is to derive.
- You must know how to derive a simple or simple function.
- Know what a complex function is. If you do not know, you can access its website in the free encyclopedia.
- You must know the operation and concept of the chain rule. Having resorted to our guide, we have decided to include the definition and operation -in a basic way- of this rule :
- The chain rule consists in a certain way of having two functions “grouped” which we are going to define by using two fundamental steps: the use of the derivation rule for the function of the exterior part and the multiplication by the derivative from the inside
- Therefore, we must add knowledge of the derivation rule to the list. If you want to know more about it, you can access the Wikipedia website to give you an idea.
As you can see, it is essential that you have a certain level of practice and/or familiarization with the world of calculus, as it is essential in order to learn how this operation works.
Materials
- White sheets, notebook, recycled sheets… wherever you want! (taking into account that you will have to do several practice exercises)
- Pencil, pens, colors. We recommend that you use colors to highlight where you have failed in each exercise, this will help you detect your most common failures and be able to pay more attention to them for the rest of the exercises.
- Eraser.
- Sharpener.
- Calculator.
- A quiet study place, where no one bothers you.
- Have at hand all the books on algebra, calculus, mathematics, in general, those that can help you to solve the derivatives.
Attitude
- I like mathematics.
- Eager to expand and improve your knowledge.
- Patience -because it is not easy at all- at least the first times.
Once given everything you need to be able to face the resolution of complex functions through the chain rule, we will begin to soak up mates by carrying out an exercise step by step.
Applying the chain rule: step by step guide (+ exercise guide)
Since we don’t know if you are a teacher looking for exercises or a student, we have decided to include two sections in our step-by-step guide. In the first (A) you will find everything inherent to solving examples on the chain rule while in the second (B) you will find a series of exercises that can be initial for your students.
Learning to solve a complex function:
The procedure to apply the chain rule is not very complex. That is, you must be aware of everything you are doing, but, once done and repeated several times, it will be a piece of cake for you.
Generally, the process and steps that are usually used and the one currently taught by most teachers internationally is the following:
- This is the most difficult step, since the whole operation starts from here, identifying u = t (x). In fact, commonly, the biggest mistake that students make is that they fail to identify u = t(x).
- As we had anticipated in the definition of the chain rule, we will seek to find the derivative of the exterior function. This translated into mathematical language is the following g (u).
- You have already identified u = t (x) and obtained the derivative of g (u), you have already done half an operation. Now you must proceed to obtain the derivative of the function that remained inside the grouping, this taken to mathematical language is expressed as follows t (x).
- Then, you will have to perform the multiplication of the derivatives, that is, obtain the product thereof, taken to the mathematical expression would be the following g ‘(u) t'(x).
- Finally you have to substitute the u for the expression t (x).
- In theory it sounds very simple right? This will remind you of school when the teacher or math teacher explained an operation and it seemed simple to you -like everyone else- but then nobody knew how to do it well on the exam, except for a small group.
Don’t worry, the same thing won’t happen to you with the chain rule because now we will do an initial exercise:
- Exercise #1: Suppose we are told to find the derivative of the following function: f (x)= (x2 + 1)4. Let’s see how to solve it using the chain rule following step by step what is explained in the theoretical section.
- First you need to identify u = t(x). In this case it would be u = t (x) = x2 + 1.
- Now we have to proceed to look for the derivative of the exterior function or, as we explained it to you above, g (u). In this case it would be the following: g (u) = u4, the derivative is g ‘(u) = 4u3.
- Then you will have to obtain the derivative of the interior which is the following: t'(x) = 2x.
- Now…. We will let you obtain the product of the following operation g ‘(u) t'(x) and after that result substitute u for the expression t (x).
You should know that these are the simplest type of functions within this world of derivatives. The difficulty begins to increase as different operations or mathematical products are added.
You can use the chain rule to get functions that contain the following:
- Fractions: it is simple because it is not necessary that the whole process be disrupted or that you have to do others. In fact, there are those who prefer to work with fractions because they can be taken to their minimum expression.
- Roots: It doesn’t change the process much either but it does add a little difficulty. More than anything in cases where you find more than one root in the function and not a single root that encompasses the entire function.
- Vectors: some people are afraid of vectors and they are certainly not the easiest to do. With the chain rule you can also obtain derivatives of functions with vectors, yes, you will have to pay attention to the different axes.
- Variables: can be of two types:
- Independent: it is only one type of letter that appears in the function (Example: x).
- Two independent: it is the case in which you have more than one variable, that is, there is more than one letter (Example: x and y).
- Forms of expression: there is an alternative way to write the chain rule, that is, everything we have done before and it is called Leibniz Notation. However, it is best that you first learn how to solve derivatives of complex functions this way, and then dive into more complex terms.
The chain rule can be taken to more complex terms as exposed by an Italian mathematician named Faà di Bruno, who applied the same to the so-called derivatives of higher order. If you want to know more about this, you can enter the website dedicated to the free encyclopedia.
Exercises for students:
Whether you are a teacher or a student, you will need exercises to carry out the chain rule, so we leave you below 5 exercises to do at home or in class:
- f (x) = 8/3 (x2 – 3)2
- f(x) = (5×2 – 1)4
- f(x) = (x3 – 4×2)3
- f(x) = (x3 – 3)
- f (x) = 15/5 (x3 – 4)2
You have exercises for a while! So there’s no excuse…
Chain Rule: Tips
You have probably reached this point in the guide and you have two very contrasting positions, a first optimistic that you can achieve it and a second that is not so positive -rather negative- that makes you doubt your mathematical abilities.
It is essential that you know that this is knowledge that we should not necessarily integrate quickly, that is, you can try to assimilate it quickly but even so, this does not mean that you are an expert applying the chain rule from day one.
Tips
Chain Rule: Tips
You have probably reached this point in the guide and you have two very contrasting positions, a first optimistic that you can achieve it and a second that is not so positive -rather negative- that makes you doubt your mathematical abilities.
It is essential that you know that this is knowledge that we should not necessarily integrate quickly, that is, you can try to assimilate it quickly but even so, this does not mean that you are an expert applying the chain rule from day one.
If you want to know how to apply the chain rule correctly, you should follow the following tips:
- Have quite fresh and well-learned previous theoretical knowledge. If not, better review them.
- To practice…. To practice…. And more practice! You will not pretend to be the best without trying very hard. You need to do a lot of exercises to become an expert in the chain rule.
- It is important that you accompany all that effort and knowledge with a positive attitude so that you never give up and keep persevering… Remember: no one was born learned!
Finally, we want to thank you for having read our guide and we hope that it has helped you to expand your knowledge… Come on, you can! If you need to review knowledge, just look at the links that we recommend in this post and also review our blog.